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TMUA 2020 · Paper 2 · Question 14 of 20

TMUA 2020 Paper 2 Question 14

Sequences and series — Arithmetic series · when two partial sums agree. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2Sequences and seriesArithmetic series · when two partial sums agree4 options
An arithmetic sequence T has first term a and common difference d, where a and d are non-zero integers.

Property P is:

For some positive integer m, the sum of the first m terms of the sequence is equal to the sum of the first 2m terms of the sequence.

For example, when a = 11 and d = −2, the sequence T has property P, because 11 + 9 + 7 + 5 = 11 + 9 + 7 + 5 + 3 + 1 + (−1) + (−3) i.e. the sum of the first 4 terms equals the sum of the first 8 terms.

Which of the following statements is/are true?

I   For T to have property P, it is sufficient that ad < 0.
II  For T to have property P, it is necessary that d is even.

  1. Aneither of them
  2. BI only
  3. CII only
  4. DI and II
Show the answer and worked solution
answer · A
  1. Aneither of them
  2. BI only
  3. CII only
  4. DI and II
Set up the condition once. With Sm = m2(2a + (m−1)d), the equation Sm = S2m simplifies (after dividing by m, which is non-zero) to 2a + (3m−1)d = 0. So property P holds exactly when m = 13(1 − 2ad) is a positive integer. For I, take a = 2, d = −1: here ad < 0, but the equation gives 4 − (3m−1) = 0, so m = 53, not an integer, and P fails. For II, take a = −1, d = 1: the equation gives −2 + 3m − 1 = 0, so m = 1, and indeed −1 = −1 + 0 — property P holds with d odd. Both statements fail.